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This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For $k = 0$ one gets 2 dimensional string world sheet, For $k = 1$ one gets 10 spacetime dimensions and for $k = 3$ one gets 26 dimensions of bosonic string theory. For $k = 2$ we get 18. I don't know whether it has any relevance or not in ST. Also the number 24, which can be thought of as number of dimensions perpendicular to 2 dimensional string world sheet in bosonic ST, is the largest number for which the sum of squares up to 24 is itself a square. $(1^2 + 2^2 + ..+24^2 = 70^2)$

My question is, is it a mere coincidence or something deeper than that?

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    $\begingroup$ Excellent observations. It's indeed natural to count the transverse coordinates only - the number of physical "oscillator" degrees of freedom - and those transverse dimensionalities are multiples of eight. This is linked to the fact that the dimension of a spin field is $1/16$ for a single dimension and one needs dimensions that are integral or half-integral. In theories with spacetime fermions, it's also linked to the Bott periodicity - if the difference between spatial and temporal dimensions is a multiple of eight, there are real chiral spinor representations. $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 16:16
  • $\begingroup$ Also, the number 24 for the transverse dimension of the bosonic string appears because one needs to get the right critical dimension, and the zero-point energy with the single excitation has to vanish: $(D-2)(-1/12)/2+1=0$. This is solved exactly for $D=26$; $(-1/12)$ arose as the sum of positive integers or $\zeta(-1)$. Incredibly enough, even the seemingly numerological observation with $70^2$ is actually used "somewhere" in string theory - one compactified on the Leech lattice. The identity guarantees that a null vector is null. $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 16:19
  • $\begingroup$ Under the comment by Dr Harvey, I link to a paper where a string theory compactification on the Leech lattice is actually used to explain even more fascinating numerological accidents - the "monstrous moonshine" linking some properties of the monster group, the largest finite sporadic group, to some properties of number theory and complex calculus, previously totally unrelated part of maths. See en.wikipedia.org/wiki/Monstrous_moonshine - In Monstrous Moonshine, numbers as high as 196,883+1 appear at 2 places and it was a complete mystery why! String theory has demystified this fact. $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 16:25
  • $\begingroup$ Dear @sb1, as far as I know, there is no important string theory in $d=18$, much like there's no important string theory in $d=34$ and higher numbers of the form $8k+2$. The numerology here is much deeper than just "all numbers of the form $8k+2$": this template for a number is just one aspect of the story but it's not true that all numbers of this form are equally important as dimensionalities. $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 16:27
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    $\begingroup$ @kakemonster: Numerology is to number theory as astrology is to astronomy, or alchemy is to chemistry. $\endgroup$
    – QGR
    Commented Feb 17, 2011 at 6:08

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There is definitely something deep going on, but there is not yet a deep understanding of what it is. In math the topology of the orthogonal group has a mod 8 periodicity called Bott periodicity. I think this is related to the dimensions in which one can have Majorana-Weyl spinors with Lorentzian signature which is indeed $8k+2$. So this is part of the connection and allows both the world-sheet and the spacetime for $d=2,10$ to have M-W spinors. The $26$ you get for $k=3$ doesn't have any obvious connection with spinors and supersymmetry, but there are some indirect connections related to the construction of a Vertex Operator Algebra with the Monster as its symmetry group. This involves a $Z_2$ orbifold of the bosonic string on the torus $R^{24}/\Lambda$ where $\Lambda$ is the Leech lattice. A $Z_2$ orbifold of this theory involves a twist field of dimension $24/16=3/2$ which is the dimension needed for a superconformal generator. So the fact that there are $24$ transverse dimensions does get related to world-sheet superconformal invariance. Finally, the fact you mentioned involving the sum of squares up to $24^2$ has been exploited in the math literature to give a very elegant construction of the Leech lattice starting from the Lorentzian lattice $\Pi^{25,1}$ by projecting along a null vector $(1,2, \cdots 24;70)$ which is null by the identity you quoted. I can't think of anything off the top of my head related to $k=2$ in string theory, but I'm sure there must be something.

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    $\begingroup$ Prof Harvey may be too modest here but let me mention that he is one of the 4 co-fathers of the heterotic string. And when it comes to a related compactification on the Leech lattice, see e.g. Beauty and the Beast: web.mac.com/chrisbertinato/iWeb/Physics/Seminars_files/… - This compactification of string theory actually knows about most (or all) about the largest sporadic finite group, the monster group. Fascinating and previously "impossible" connections between number theory and group theory - the "monstrous moonshine" - has been explained as a real link here. $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 16:22
  • $\begingroup$ Just a direct link to the construction of the Leech lattice where the "70 squared" identity is used: en.wikipedia.org/wiki/… $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 17:08
  • $\begingroup$ Yes, thanks @Jeff. And I am aware that they're the authors. Sorry I didn't make it clear. The final proof of the monstrous moonshine claim that won the Fields medal was found by Borcherds - just to make it clear that I acknowledge that this Gentleman has some divine abilities, too. ;-) $\endgroup$
    – Luboš Motl
    Commented Feb 15, 2011 at 17:10
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The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in $s=1$ is cancelled via a multiplication that produces the Dirichlet eta function, $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.

This function has better analiticity than Zeta. But it is alternating, $$ \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$ And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,

$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$

And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$

So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8.

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I am separating this answer from the other because it is overly speculative; mainly I wanted to list a few hints about the sequence I named in the comments.

The OP names a square sum that happens to be related to the critical dimension of space-time of the bosonic string, D-2=24. It seems natural to ask if there is some similar sequence for the critical dimension of the superstring, D-2=8. On the other hand, as I said in the other answer, for the open superstring the Zeta regularization is naturally substituted, in some cases, by the Dirichlet Eta, which happens to be an alternating sum. So it is natural for a numerologist to try the alternating square sum and, as said in the comments to the OP, it works: $$1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2= -36 = -6^2$$

The main difference, number-wise, with the non-alternating sum, is that here the solution is not unique. Still, it is the smallest non-trivial one, and all the others can be generated iteratively: the (absolute value of the) sums are the triangular square numbers, and it was observed by Colin Dickson (alt.math.recreational March 7th 2004) that such numbers obey a recurrence law $$a_{n+1}={(a_n -1)^2 \over a_{n-1}}$$ with the first two terms being the trivial $a_1=1$ and the above $a_2=36$. For more info, see the OEIS sequences A001110 and A001108. Note that the sign in the actual solution depends on the number of terms in the sequence, alternating itself, so that actually the sum is $\sigma_n= (-1)^{n-1} a_n$

A way to produce the alternating solutions is to solve the Pell equation, and then the sums are also produced from Pell numbers, via $a_n= P_n (P_n+P_{n-1})$. This could be interesting because the root of the non-alternating series, $70$, is a Pell number itself, the 6th, and the next Pell number, $70+(70+29)=169$, is the only Pell number that is an exact square (and the only one that is an exact power).

In A001108, Mohamed Bouhamida mentions some periodicities and some mod 8 relationships for the series. Also, the page http://www.cut-the-knot.org/do_you_know/triSquare.shtml gives some hints on some eight factors appearing in a particular subsequence of square triangular numbers: "Eight triangles increased by unity produce a square". If these factors can be related to the 8-periodicities of Bott theory or theta functions, I can not tell.

EDIT: of course, the use of triangular numbers can be telling us that all the business of alternating series is just a decoy: pairing the terms, we can reduce $ (m+1)^2 - m^2 = 2 m + 1 = (m+1) + m $ the alternating squared series to the non-alternating non-squared series and then simply $$1+2+3+4+5+6+7+8= 36 = 6^2$$ But the goal is to keep at least a formal likeliness with the OP series.

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I don't know about the particular example that you mention, but there are certainly some interconnections with special numbers in mathematics and in string theory/supersymmetry.

One worked out example is the connection of possible dimensions of supersymmetry in dimensions 3, 4, 6 or 10 which is connected to the existence of normed division algebras in dimensions 1, 2, 4 and 8. For more details see

  • John C. Baez, John Huerta: Division Algebras and Supersymmetry I (arXiv)

and related work about higher gauge theory.

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  • $\begingroup$ Sorry for being off-topic, but I thought the relationship I mention interesting enough in the given context in its own right. $\endgroup$
    – Tim van Beek
    Commented Feb 15, 2011 at 18:39
  • $\begingroup$ it is one of the most important "numerlogical" facts in these considerations, so +1 $\endgroup$
    – user346
    Commented Feb 16, 2011 at 5:24

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